Related papers: Indeterminate Jacobi operators

In this paper we construct Donoghue $m$-functions for the Jacobi differential operator in $L^2\big((-1,1); (1-x)^{\alpha} (1+x)^{\beta} dx\big)$, associated to the differential expression \begin{align*} \begin{split} \tau_{\alpha,\beta} = -…

We present in this work a proof of the exponential dichotomy for dynamically defined matrix-valued Jacobi operators in $(\mathbb{C}^{l})^{\mathbb{Z}}$, given for each $\omega \in \Omega$ by the law $[H_{\omega} \textbf{u}]_{n} := D(T^{n -…

In this paper we study the truncated operator trigonometric moment problem. All solutions of the moment problem are described by a Nevanlinna-type parameterization. In the case of moments acting in a separable Hilbert space, the matrices of…

Jacobi's $\theta$ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of $\theta(z,\tau)$, for $z, \tau$ verifying certain conditions, with precision $P$ in $O(\mathcal{M}(P)…

We study the spectral properties of bounded and unbounded Jacobi matrices whose entries are bounded operators on a complex Hilbert space. In particular, we formulate conditions assuring that the spectrum of the studied operators is…

We prove that a solution of the Schr\"odinger-type equation $\mathrm{i}\partial_t u= Hu$, where $H$ is a Jacobi operator with asymptotically constant coefficients, cannot decay too fast at two different times unless it is trivial.

We consider operators acting on a Hilbert space that can be written as the sum of a shift and a diagonal operator and determine when the operator is hyponormal. The condition is presented in terms of the norm of an explicit block Jacobi…

We obtain a finite-sum representation for the general solution of the Jacobi second-order difference equation D(p(n-1)Du(n-1))+q(n)u(n)=l r(n)u(n) in terms of a nonvanishing solution corresponding to some fixed value of the spectral…

A semi-infinite weighted Hankel matrix with entries defined in terms of basic hypergeometric series is explicitly diagonalized as an operator on $\ell^{2}(\mathbb{N}_{0})$. The approach uses the fact that the operator commutes with a…

We consider a periodic Jacobi operator $H$ with finitely supported perturbations on ${\Bbb Z}.$ We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the scattering data: the inverse of…

k:th power (amplitude-)squeezed states are defined as the normalized states giving equality in the Schroedinger-Robertson uncertainty relation for the real and imaginary parts of the k:th power of the one-mode annihilation operator.…

The complex moment sequence m(P) is assigned to a univalent polynomial P by the Cauchy transform of the P(D), where D is the unit disk. We establish the representation of the Jacobian det dm(P) in terms of roots of the derivative P'.…

We provide a full classification scheme for exceptional Jacobi operators and polynomials. The classification contains six degeneracy classes according to whether $\alpha,\beta$ or $\alpha\pm\beta$ assume integer values. Exceptional Jacobi…

Four new examples of explicitly diagonalizable Hankel matrices depending on a parameter $k\in(0,1)$ are presented. The Hankel matrices are regarded as matrix operators on the Hilbert space $\ell^{2}(\mathbb{N}_{0})$ and the solution of the…

We investigate the differential equation for the Jacobi-type polynomials which are orthogonal on the interval $[-1,1]$ with respect to the classical Jacobi measure and an additional point mass at one endpoint. This scale of higher-order…

Our main result asserts that a certain natural non-linear operator on Jacobi matrices built by a hyperbolic polynomial with real Julia set is a contraction in operator norm if the polynomial is sufficiently hyperbolic. This allows us to get…

We study some classes of symmetric operators for the discrete series representations of the quantum algebra U_q(su_{1,1}), which may serve as Hamiltonians of various physical systems. The problem of diagonalization of these operators…

We consider the inverse dynamical problem for the dynamical system with discrete time associated with the semi-infinite Jacobi matrix. We solve the inverse problem for such a system and answer a question on the characterization of the…

We prove local solvability for large classes of operators of the form $$ L=\sum_{j,k=1}^{2n}a_{jk}V_jV_k+i\alpha U,$$ where the $V_j$ are left-invariant vector fields on the Heisenberg group satisfying the commutation relations…

We study the trace class perturbations of the whole-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we refine the Lieb--Thirring…